91Ó°ÊÓ

The time \(t\) (in seconds) it takes for sound to travel 1 kilometer can be modeled by $$ t=\frac{1000}{0.6 T+331} $$ where \(T\) is the air temperature (in degrees Celsius). a. You are 1 kilometer from a lightning strike. You hear the thunder \(2.9\) seconds later. Use a graph to find the approximate air temperature. b. Find the average rate of change in the time it takes sound to travel 1 kilometer as the air temperature increases from \(0^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C}\).

A business is studying the cost to remove a pollutant from the ground at its site. The function \(y=\frac{15 x}{1.1-x}\) models the estimated cost \(y\) (in thousands of dollars) to remove \(x\) percent (expressed as a decimal) of the pollutant. a. Graph the function. Describe a reasonable domain and range. b. How much does it cost to remove \(20 \%\) of the pollutant? \(40 \%\) of the pollutant? \(80 \%\) of the pollutant? Does doubling the percentage of the pollutant removed double the cost? Explain.

You plan a trip that involves a 40 -mile bus ride and a train ride. The entire trip is 140 miles. The time (in hours) the bus travels is \(y_1=\frac{40}{x}\), where \(x\) is the average speed (in miles per hour) of the bus. The time (in hours) the train travels is \(y_2=\frac{100}{x+30}\). Write and simplify a model that shows the total time \(y\) of the trip.

Find two rational functions f and g that have the stated product and quotient. $$ (f g)(x)=x^2,\left(\frac{f}{g}\right)(x)=\frac{(x-1)^2}{(x+2)^2} $$

Your friend claims that the least common multiple of two numbers is always greater than each of the numbers. Is your friend correct? Justify your answer.

Golden rectangles are rectangles for which the ratio of the width \(w\) to th length \(\ell\) is equal to the ratio of \(\ell\) to \(\ell+w\). The ratio of the length to the width for these rectangles is called the golden ratio. Find the value of the golden ratio using a rectangle with a width of 1 unit.

You borrow \(P\) dollars to buy a car and agree to repay the loan over \(t\) years at a monthly interest rate of \(i\) (expressed as a decimal). Your monthly payment \(M\) is given by either formula below. $$ M=\frac{P i}{1-\left(\frac{1}{1+i}\right)^{12 t}} \quad \text { or } \quad M=\frac{P i(1+i)^{12 t}}{(1+i)^{12 t}-1} $$ a. Show that the formulas are equivalent by simplifying the first formula. b. Find your monthly payment when you borrow \(\$ 15,500\) at a monthly interest rate of \(0.5 \%\) and repay the loan over 4 years.

Your friend claims it is possible for a rational function to have two vertical asymptotes. Is your friend correct? Justify your answer.

Is it possible to write two rational functions whose sum is a quadratic function? Justify your answer.

In what line(s) is the graph of \(y=\frac{1}{x}\) symmetric? What does this symmetry tell you about the inverse of the function \(f(x)=\frac{1}{x}\) ?